Recent advances in artificial intelligence (AI), especially in generative language modelling, hold the promise of transforming government. Given the advanced capabilities of new AI systems, it is critical that these are embedded using standard operational procedures, clear epistemic criteria, and behave in alignment with the normative expectations of society. Scholars in multiple domains have subsequently begun to conceptualize the different forms that AI applications may take, highlighting both their potential benefits and pitfalls. However, the literature remains fragmented, with researchers in social science disciplines like public administration and political science, and the fast-moving fields of AI, ML, and robotics, all developing concepts in relative isolation. Although there are calls to formalize the emerging study of AI in government, a balanced account that captures the full depth of theoretical perspectives needed to understand the consequences of embedding AI into a public sector context is lacking. Here, we unify efforts across social and technical disciplines by first conducting an integrative literature review to identify and cluster 69 key terms that frequently co-occur in the multidisciplinary study of AI. We then build on the results of this bibliometric analysis to propose three new multifaceted concepts for understanding and analysing AI-based systems for government (AI-GOV) in a more unified way: (1) operational fitness, (2) epistemic alignment, and (3) normative divergence. Finally, we put these concepts to work by using them as dimensions in a conceptual typology of AI-GOV and connecting each with emerging AI technical measurement standards to encourage operationalization, foster cross-disciplinary dialogue, and stimulate debate among those aiming to rethink government with AI.